Generally the word kafir means 'unbeliever' and it is not meant as a derogatory label (unless it is used against Muslims by another Muslim). Of course, there are different levels of Kufur (disbelief) so although a person may be correctly identified as a kafir, that person is not kafir in the strongest sense of the word. Absolute kufur is the denial of God. Here are some examples of kufur:
(1) A person is known as a Kafir if they are disbelievers in Allah, in His Oneness, and in His final Messenger, Muhammad (SAW). (E.g. An apostate from Islam)
(2) A 'Muslim' who disbelieves in a necessary tenant of Islam like the 5 pillars or some other things is known as a disbeliever.
(3) An originally non-Muslim person who denies the religion of Islam while knowing in his heart that it is true.
(4) A disbeliever in Islam only because they originally follow another (monotheistic) faith.
(5) A person who follows another (monotheistic) faith and knows nothing about Islam.
(6) A person who believes in many gods (polytheist).
(7) An agnostic who is a monotheist is a lesser kafir than a polytheist.
(8) An Atheist - ultimate kufur.
Generally a Christian or Jew is not identified as a kafir but they are known respectfully as Ahlul kitab - People of the Book(s), and Islam considers their religions valid but once they find Islam they are expected to follow the updated covenant sent by God.
وَإِذْ قُلْنَا لِلْمَلاَئِكَةِ اسْجُدُواْ لآدَمَ فَسَجَدُواْ إِلاَّ إِبْلِيسَ أَبَى وَاسْتَكْبَرَ وَكَانَ مِنَ الْكَافِرِينَ
And behold, We said to the angels: "Bow down to Adam" and they bowed down. Not so Iblis: he refused and was haughty: He was of those who reject Faith. [Quran 2.34]
Notice the use of the word 'kafireen' in the Arabic. This is one of the reasons why one must learn Arabic to understand the full meaning of the Quran, the words of Allah. Kafir is derived from the word meaning 'to cover' and Kafir also means 'disbeliever in God'. But in this case, Iblees (Satan) is fully aware of Gods' existence and power yet he is still described as a kafir! This goes to show that a person can be a kafir even if he believes in a God. In other words, just because someone believes there is a God, that someone can still be a kafir and be (technically) condemned to hell. Therefore, a kafir can be more correctly defined as someone who 'rejects submission to God' even though he is fully aware of His existence, as well as meaning a person who is a true Atheist - one who 'rejects the existence of God'.